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  • QUASI-ANOSOV DIFFEOMORPHISMS OF 3-MANIFOLDS

    T. FISHER, M. RODRIGUEZ HERTZ

    Abstract. In 1969, Hirsch posed the following problem: given a dif- feomorphism f : N → N , and a compact invariant hyperbolic set Λ of f , describe the topology of Λ and the dynamics of f restricted to Λ. We solve the problem where Λ = M3 is a closed 3-manifold: if M3 is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions.

    The dynamics of the diffeomorphisms restricted to M3, called quasi- Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions.

    1. Introduction

    This paper deals with hyperbolic sub-dynamics. It is related to a problem posed by M. Hirsch, around 1969: given a diffeomorphism f : N → N , and a compact invariant hyperbolic set Λ of f , describe the topology of Λ and the dynamics of f restricted to Λ. Hirsch asked, in particular, whether the fact that Λ were a manifold M would imply that the restriction of f to M is an Anosov diffeomorphism [11]. However, in 1976, Franks and Robinson gave an example of a non-Anosov hyperbolic sub-dynamics in the connected sum of two T3 [3] (see below). There are also examples of hyperbolic sub-dynamics in non-orientable 3-manifolds, for instance, the example of Zhuzhoma and Medvedev [18]. We show here that all examples of 3-manifolds that are hyperbolic invariant sets are, in fact, finite connected sums of the examples above and handles S2 × S1 (see definitions in §3 and §5)

    Theorem 1.1. Let f : N → N be a diffeomorphism, and let M ⊂ N be a hyperbolic invariant set for f such that M is a closed orientable 3-manifold. Then the Kneser-Milnor prime decomposition of M is

    M = T1# . . .#Tk#H1# . . .#Hr

    Date: May 7, 2007. 2000 Mathematics Subject Classification. 37D05; 37D20. Key words and phrases. Dynamical Systems, Hyperbolic Set, Robustly Expansive,

    Quasi-Anosov. Partially supported by NSF Grant #DMS0240049, Fondo Clemente Estable 9021 and

    PDT.

  • 2 T. FISHER, M. RODRIGUEZ HERTZ

    the connected sum of k ≥ 1 tori Ti = T3 and r ≥ 0 handles Hj = S2 × S1. In case M is non-orientable, then M decomposes as

    M = T̃1# . . .#T̃k#H1# . . .#Hr

    the connected sum of k ≥ 1 tori quotiented by involutions T̃i = T3|θi and r handles Hj = S2 × S1.

    In 1976, Mañé obtained the following characterization [15] (see also The- orem 3.3): g : M → M is the restriction of another diffeomorphism to a hyperbolic set M that is a closed manifold, if and only if g is quasi-Anosov; that is, if it satisfies Axiom A and all intersections of stable and unstable manifolds are quasi-transversal, i.e.:

    (1.1) TxW s(x) ∩ TxW

    u(x) = {0} ∀x ∈ M

    The Franks-Robinson’s example of a non-Anosov quasi-Anosov diffeo- morphism is essentially as follows: they consider a hyperbolic linear au- tomorphism of a torus T1 with only one fixed point, and its inverse in another torus T2. They produce appropriate deformations on each torus (DA-diffeomorphisms) around their respective fixed points. Then they cut suitable neighborhoods containing these fixed points, and carefully glue together along their boundary so that the stable and unstable foliations intersect quasi-transversally. This is a quasi-Anosov diffeomorphism in the connected sum of T1 and T2, and hence T1#T2 is a compact invariant hyper- bolic set of some diffeomorphism. The non-orientable example by Medvedev and Zhuzhoma [18] is similar to Franks and Robinson’s, but they perform a quotient of each Ti by an involution before gluing them together.

    The second part of this work, a classification of the dynamics of quasi- Anosov diffeomorphisms of 3-manifolds, shows that all examples are, in fact, connected sums of the basic examples above:

    Theorem 1.2. Let g : M → M be a quasi-Anosov diffeomorphism of a closed 3-manifold M . Then

    (1) The non-wandering set Ω(g) of g consists of a finite number of codimension-one expanding attractors, codimension-one shrinking re- pellers and hyperbolic periodic points.

    (2) For each attractor Λ in Ω(g), there exist a hyperbolic toral automor- phism A with stable index one, a finite set Q of A-periodic points, and a linear involution θ of T3 fixing Q such that the restriction of g to its basin of attraction W s(Λ) is topologically conjugate to a DA-diffeomorphism fAQ on the punctured torus T

    3 −Q quotiented by θ. In case M is an orientable manifold, θ is the identity map. An analogous result holds for the repellers of Ω(g).

  • 3

    Item (2) above is actually a consequence of item (1), as it was shown by Plykin in [20, 21], see also [6] and [7]. A statement of the result can be found in Theorem 4.3 in this work. The proof of Theorem 1.2 is in §4. Theorem 1.2, in fact, implies Theorem 1.1. This is proved in §5.

    Let us see how a handle S2×S1 could appear in the prime decomposition of M : Consider a linear automorphism of a torus T1, and its inverse in a torus T2, as in Franks-Robinson’s example. Then, instead of exploding a fixed point, one explodes and cuts around an orbit of period 2 in T1 and in T2. The rest of the construction is very similar, gluing carefully as in that example to obtain a quasi-Anosov dynamics. This gives the connected sum of two tori and a handle. Explanation and details can be found in §5.

    Let us also mention that in a previous work [22] it was shown there exist a codimension-one expanding attractor and a codimension-one shrinking repeller if g is a quasi-Anosov diffeomorphism of a 3-manifold that is not Anosov. The fact that only T3 can be an invariant subset of any known Anosov system was already shown by A. Zeghib [29]. In that case, the dynamics is Anosov. See also [2] and [16].

    This work is also related to a work by Grines and Zhuzhoma [8]. There they prove that if an n-manifold supports a structurally stable diffeomor- phism with a codimension-one expanding attractor, then it is homotopy equivalent to Tn, and homeomorphic to Tn if n )= 4. In a certain sense, the results deal with complementary extreme situations in the Axiom A world: Grines-Zhuzhoma result deals with structurally stable diffeomor- phisms, which are Axiom A satisfying the strong transversality condition. This means that all x, y in the non-wandering set satisfy at their points z of intersection: TzW s(x) ! TzW u(y). In particular,

    dim Esx + dim E u y ≥ n

    In our case, we deal with quasi-Anosov diffeomorphisms, which are Ax- iom A satisfying equality (1.1). In particular, for x, y, z as above, we have TzW

    s(x) ∩ TzW u(y) = {0}, so:

    dim Esx + dim E u y ≤ n

    In the intersection of both situations are, naturally, the Anosov diffeomor- phisms.

    Observe that it makes sense to get a classification of the dynamical be- havior of quasi-Anosov on its non-wandering set, since quasi-Anosov are Ω- stable [15] (see also §3). They form an open set, due to quasi-transversality condition (1.1). Moreover, they are the C1-interior of expansive diffeomor- phisms, that is, they are robustly expansive [14]. However, 3-dimensional quasi-Anosov diffeomorphisms of M )= T3 are never structurally stable, so

  • 4 T. FISHER, M. RODRIGUEZ HERTZ

    they are approximated by other quasi-Anosov diffeomorphisms with differ- ent dynamical behavior, but similar asymptotic behavior (Proposition 3.2).

    Finally, in Section 7 we study quasi-Anosov diffeomorphisms in the pres- ence of partial hyperbolicity (see definitions in §7). We obtain the following result under mild assumptions on dynamical coherence:

    Theorem 1.3. If f : M3 → M3 is a quasi-Anosov diffeomorphism that is partially hyperbolic, and either Ecs or Ecu integrate to a foliation, then f is Anosov.

    Acknowledgments. We want to thank the referees for valuable comments, and Raúl Ures for suggestions. The second author is grateful to the Depart- ment of Mathematics of the University of Toronto, and specially to Mike Shub, for kind hospitality.

    2. Basic definitions

    Let us recall some basic definitions and facts: Given a diffeomorphism f : N → N , a compact invariant set Λ is a hyperbolic set for f if there is a Tf -invariant splitting of TN on Λ:

    TxN = E s x ⊕ E

    u x ∀x ∈ Λ

    such that all unit vectors vσ ∈ EσΛ, with σ = s, u satisfy

    |Tf(x)vs| < 1 < |Tf(x)vu|

    for some suitable Riemannian metric |.|. The non-wandering set of a dif- feomorphism g : M → M is denoted by Ω(g) and consists of the points x ∈ M , such that for each neighborhood U of x, the family {gn(U)}n∈Z is not pairwise disjoint. The diffeomorphism g : M → M satisfies Axiom A if Ω(g) is a hyperbolic set for g and periodic points are dense in Ω(g). The stable manifold of a point x is the set

    W s(x) = {y ∈ M : d(fn(x), fn(y)) → 0 if n → ∞}

    where d(., .) is the induced metric; the unstable manifold W u(x) is defined analogously for n → −∞. If g satisfies Axiom A, then W s(x) and W u(x) are immersed manifolds for each x ∈ M (see for instance [25]). Also, if dσ is the intrinsic metric of the invariant manifold W σ(x), for σ = s, u, one has constants C, ε > 0 and 0 < λ < 1 such that, for instance, if y ∈ W s(x), and ds(x, y) ≤ ε for some small ε > 0 then

    (2.2) ds(f n(x), fn(y)) ≤ Cλnds(x, y) ∀n ≥ 0

    an analogous bound holds for the unstable manifold.

  • 5

    Due to the Spectral Decomposition Theorem of Smale [26], if g is Axiom A, then Ω(g) can be decomposed into disjoint compact invariant sets, called basic sets: